Existing studies on preference optimization (PO) have been focused on constructing pairwise preference data following simple heuristics, such as maximizing the margin between chosen and rejected responses based on human (or AI) ratings. In this work, we develop a novel PO framework that provides theoretical guidance to effectively sample rejected responses. To achieve this, we formulate PO as minimizing the negative log-likelihood (NLL) of a probability model and propose a sampling-based solution to estimate its normalization constant via contrastive divergence. We show that these estimative samples can act as rejected responses in PO. Leveraging the connection established between PO and NLL estimation, we propose a novel PO algorithm, called Monte-Carlo-based PO (MC-PO), that applies a MC kernel to sample *hard negatives* w.r.t.~the log-likelihood of the target policy. Intuitively, these hard negatives represent the rejected samples that are most difficult for the current policy to differentiate. We show that MC-PO outperforms existing SOTA baselines on popular alignment benchmarks.

In Markov decision processes (MDPs), quantile risk measures such as Value-at-Risk are a standard metric for modeling RL agents’ preferences for certain outcomes. This paper proposes a new Q-learning algorithm for quantile optimization in MDPs with strong convergence and performance guarantees. The algorithm leverages a new, simple dynamic program (DP) decomposition for quantile MDPs. Compared with prior work, our DP decomposition requires neither known transition probabilities nor solving complex saddle point equations and serves as a suitable foundation for other model-free RL algorithms. Our numerical results in tabular domains show that our Q-learning algorithm converges to its DP variant and outperforms earlier algorithms.

Conservative Contextual Bandits (CCBs) address safety in sequential decision making by requiring that an agent's policy, along with minimizing regret, also satisfies a safety constraint: the performance is not worse than a baseline policy (e.g., the policy that the company has in production) by more than $(1+\alpha)$ factor. Prior work developed UCB-style algorithms for this problem in the multi-armed (Wu et al., 2016) and contextual linear (Kazerouni et al., 2017) settings. However, in practice the cost of the arms is often a non-linear function, and therefore existing UCB algorithms are ineffective in such settings. In this paper, we consider CCBs beyond the linear case and develop two algorithms $\mathtt{C\text{-}SquareCB}$ and $\mathtt{C\text{-}FastCB}$, using Inverse Gap Weighting (IGW) based exploration and an online regression oracle. We show that the safety constraint is satisfied in high probability and that the regret for $\mathtt{C\text{-}SquareCB}$ is sub-linear in horizon $T$, while the the regret for $\mathtt{C\text{-}FastCB}$ is first-order and is sub-linear in $L^*$, the cumulative loss of the optimal policy. Subsequently, we use a neural network for function approximation and online gradient descent as the regression oracle to provide $\tilde{\mathcal{O}}\big(\sqrt{KT} + K/\alpha\big) $ and $\tilde{\mathcal{O}}\big(\sqrt{KL^*} + K (1 + 1/\alpha)\big)$ regret bounds respectively. Finally, we demonstrate the efficacy of our algorithms on real world data, and show that they significantly outperform the existing baseline while maintaining the performance guarantee.

Recent trends in test-time scaling for reasoning models (e.g., OpenAI o1, DeepSeek R1) have led to a popular belief that extending thinking traces using prompts like “Wait” or “Let me rethink” can improve performance. This raises a natural question: Does thinking more at test-time truly lead to better reasoning? To answer this question, we perform a detailed empirical study across models and benchmarks, which reveals a consistent pattern of initial performance improvements from additional thinking followed by a decline, due to "overthinking". To understand this non-monotonic trend, we consider a simple probabilistic model, which reveals that additional thinking increases output variance—creating an illusion of improved reasoning while ultimately undermining precision. Thus, observed gains from "more thinking" are not true indicators of improved reasoning, but artifacts stemming from the connection between model uncertainty and evaluation metric. This suggests that test-time scaling through extended thinking is not an effective way to utilize the inference thinking budget. Recognizing these limitations, we introduce an alternative test-time scaling approach, parallel thinking, inspired by Best-of-N sampling. Our method generates multiple independent reasoning paths within the same inference budget and selects the most consistent response via majority vote, achieving up to 20% higher accuracy compared to extended thinking. This provides a simple yet effective mechanism for test-time scaling of reasoning models.

We study contextual bandits in the presence of a stage-wise constraint when the constraint must be satisfied both with high probability and in expectation. We start with the linear case where both the reward function and the stage-wise constraint (cost function) are linear. In each of the high probability and in expectation settings, we propose an upper-confidence bound algorithm for the problem and prove a $T$-round regret bound for it. We also prove a lower-bound for this constrained problem, show how our algorithms and analyses can be extended to multiple constraints, and provide simulations to validate our theoretical results. In the high probability setting, we describe the minimum requirements for the action set for our algorithm to be tractable. In the setting that the constraint is in expectation, we specialize our results to multi-armed bandits and propose a computationally efficient algorithm for this setting with regret analysis. Finally, we extend our results to the case where the reward and cost functions are both non-linear. We propose an algorithm for this case and prove a regret bound for it that characterize the function class complexity by the eluder dimension.

Fine-tuning text-to-image models with reward functions trained on human feedback data has proven effective for aligning model behavior with human intent. However, excessive optimization with such reward models, which serve as mere proxy objectives, can compromise the performance of fine-tuned models, a phenomenon known as reward overoptimization. To investigate this issue in depth, we introduce the Text-Image Alignment Assessment (TIA2) benchmark, which comprises a diverse collection of text prompts, images, and human annotations. Our evaluation of several state-of-the-art reward models on this benchmark reveals their frequent misalignment with human assessment. We empirically demonstrate that overoptimization occurs notably when a poorly aligned reward model is used as the fine-tuning objective. To address this, we propose TextNorm, a simple method that enhances alignment based on a measure of reward model confidence estimated across a set of semantically contrastive text prompts. We demonstrate that incorporating the confidence-calibrated rewards in fine-tuning effectively reduces overoptimization, resulting in twice as many wins in human evaluation for text-image alignment compared against the baseline reward models.

We propose and theoretically analyze an approach for planning with an approximate model in reinforcement learning that can reduce the adverse impact of model error. If the model is accurate enough, it accelerates the convergence to the true value function too. One of its key components is the MaxEnt Model Correction (MoCo) procedure that corrects the model’s next-state distributions based on a Maximum Entropy density estimation formulation. Based on MoCo, we introduce the Model Correcting Value Iteration (MoCoVI) algorithm, and its sampled-based variant MoCoDyna. We show that MoCoVI and MoCoDyna’s convergence can be much faster than the conventional model-free algorithms. Unlike traditional model-based algorithms, MoCoVI and MoCoDyna effectively utilize an approximate model and still converge to the correct value function.

We study how to make decisions that minimize Bayesian regret in offline linear bandits. Prior work suggests that one must take actions with maximum lower confidence bound (LCB) on their reward. We argue that reliance on LCB is inherently flawed in this setting and propose a new algorithm that directly minimizes upper-bounds on the Bayesian regret using efficient conic optimization solvers. Our bounds build heavily on new connections to monetary risk measures. Proving a matching lower-bound, we show that our upper-bounds are tight, and by minimizing them we are guaranteed to outperform the LCB approach. Our numerical results on synthetic domains confirm that our approach is superior to maximizing LCB.

Interactive recommender systems have emerged as a promising paradigm to overcome the limitations of the primitive user feedback used by traditional recommender systems (e.g., clicks, item consumption, ratings). They allow users to express intent, preferences, constraints, and contexts in a richer fashion, often using natural language (including faceted search and dialogue). Yet more research is needed to find the most effective ways to use this feedback. One challenge is inferring a user’s semantic intent from the open-ended terms or attributes often used to describe a desired item. This is critical for recommender systems that wish to support users in their everyday, intuitive use of natural language to refine recommendation results. Leveraging concept activation vectors (CAVs) [ 26 ], a recently developed approach for model interpretability in machine learning, we develop a framework to learn a representation that captures the semantics of such attributes and connects them to user preferences and behaviors in recommender systems. One novel feature of our approach is its ability to distinguish objective and subjective attributes (both subjectivity of degree and of sense ) and associate different senses of subjective attributes with different users. We demonstrate on both synthetic and real-world datasets that our CAV representation not only accurately interprets users’ subjective semantics but also can be used to improve recommendations through interactive item critiquing .

We develop a meta-learning framework for simple regret minimization in bandits. In this framework, a learning agent interacts with a sequence of bandit tasks, which are sampled i.i.d. from an unknown prior distribution, and learns its meta-parameters to perform better on future tasks. We propose the first Bayesian and frequentist meta-learning algorithms for this setting. The Bayesian algorithm has access to a prior distribution over the meta-parameters and its meta simple regret over m bandit tasks with horizon n is mere O(m / √n). On the other hand, the meta simple regret of the frequentist algorithm is O(n√m + m/ √n). While its regret is worse, the frequentist algorithm is more general because it does not need a prior distribution over the meta-parameters. It can also be analyzed in more settings. We instantiate our algorithms for several classes of bandit problems. Our algorithms are general and we complement our theory by evaluating them empirically in several environments.

In reinforcement learning applications, we often want to accurately estimate the return of several policies of interest. We study this problem, multiple-policy high-confidence policy evaluation, where the goal is to estimate the return of all given target policies up to a desired accuracy with as few samples as possible. The natural approaches to this problem, i.e., evaluating each policy separately or estimating a model of the MDP, do not take into account the similarities between target policies and scale with the number of policies to evaluate or the size of the MDP, respectively. We present an alternative approach based on reusing samples from on-policy Monte-Carlo estimators and show that it is more sample-efficient in favorable cases. Specifically, we provide guarantees in terms of a notion of overlap of the set of target policies and shed light on when such an approach is indeed beneficial compared to existing methods.

Risk-averse Markov Decision Processes (MDPs) have optimal policies that achieve high returns with low variability, but these MDPs are often difficult to solve. Only a few practical risk-averse objectives admit a dynamic programming (DP) formulation, which is the mainstay of most MDP and RL algorithms. We derive a new DP formulation for discounted risk-averse MDPs with Entropic Risk Measure (ERM) and Entropic Value at Risk (EVaR) objectives. Our DP formulation for ERM, which is possible because of our novel definition of value function with time-dependent risk levels, can approximate optimal policies in a time that is polynomial in the approximation error. We then use the ERM algorithm to optimize the EVaR objective in polynomial time using an optimized discretization scheme. Our numerical results show the viability of our formulations and algorithms in discounted MDPs.

Many practical problems involve solving similar tasks. In recommender systems, the tasks can be users with similar preferences; in search engines, the tasks can be items with similar affinities. To learn statistically efficiently, the tasks can be organized in a hierarchy, where the task affinity is captured using an unknown latent parameter. We study the problem of off-policy learning for similar tasks from logged bandit feedback. To solve the problem, we propose a hierarchical off-policy optimization algorithm HierOPO. The key idea is to estimate the task parameters using the hierarchy and then act pessimistically with respect to them. To analyze the algorithm, we develop novel Bayesian error bounds. Our bounds are the first in off-policy learning that improve with a more informative prior and capture statistical gains due to hierarchical models. Therefore, they are of a general interest. HierOPO also performs well in practice. Our experiments demonstrate the benefits of using the hierarchy over solving each task independently.

Reinforcement learning (RL) has shown great promise for developing agents for dialogue management (DM) that are non-myopic, conduct rich conversations, and maximize overall user satisfaction. Despite the advancements in RL and language models (LMs), employing RL to drive conversational chatbots still poses significant challenges. A primary issue stems from RL’s dependency on online exploration for effective learning, a process that can be costly. Moreover, engaging in online interactions with humans during the training phase can raise safety concerns, as the LM can potentially generate unwanted outputs. This issue is exacerbated by the combinatorial action spaces facing these algorithms, as most LM agents generate responses at the word level. We develop various RL algorithms, specialized in dialogue planning, that leverage recent Mixture-of-Expert Language Models (MoE-LMs)---models that capture diverse semantics, generate utterances reflecting different intents, and are amenable for multi-turn DM. By exploiting the MoE-LM structure, our methods significantly reduce the size of the action space and improve the efficacy of RL-based DM. We evaluate our methods in open-domain dialogue to demonstrate their effectiveness with respect to the diversity of intent in generated utterances and overall DM performance.

Optimizing static risk-averse objectives in Markov decision processes is difficult because they do not admit standard dynamic programming equations common in Reinforcement Learning (RL) algorithms. Dynamic programming decompositions that augment the state space with discrete risk levels have recently gained popularity in the RL community. Prior work has shown that these decompositions are optimal when the risk level is discretized sufficiently. However, we show that these popular decompositions for Conditional-Value-at-Risk (CVaR) and Entropic-Value-at-Risk (EVaR) are inherently suboptimal regardless of the discretization level. In particular, we show that a saddle point property assumed to hold in prior literature may be violated. However, a decomposition does hold for Value-at-Risk and our proof demonstrates how this risk measure differs from CVaR and EVaR. Our findings are significant because risk-averse algorithms are used in high-stake environments, making their correctness much more critical.

We prove that, for finite-arm bandits with linear function approximation, the global convergence of policy gradient (PG) methods depends on inter-related properties between the policy update and the representation. textcolor{blue}{First}, we establish a few key observations that frame the study: \textbf{(i)} Global convergence can be achieved under linear function approximation without policy or reward realizability, both for the standard Softmax PG and natural policy gradient (NPG). \textbf{(ii)} Approximation error is not a key quantity for characterizing global convergence in either algorithm. \textbf{(iii)} The conditions on the representation that imply global convergence are different between these two algorithms. Overall, these observations call into question approximation error as an appropriate quantity for characterizing the global convergence of PG methods under linear function approximation. \textcolor{blue}{Second}, motivated by these observations, we establish new general results: \textbf{(i)} NPG with linear function approximation achieves global convergence \emph{if and only if} the projection of the reward onto the representable space preserves the optimal action's rank, a quantity that is not strongly related to approximation error. \textbf{(ii)} The global convergence of Softmax PG occurs if the representation satisfies a non-domination condition and can preserve the ranking of rewards, which goes well beyond policy or reward realizability. We provide experimental results to support these theoretical findings.

Meta-, multi-task, and federated learning can be all viewed as solving similar tasks, drawn from a distribution that reflects task similarities. We provide a unified view of all these problems, as learning to act in a hierarchical Bayesian bandit. We propose and analyze a natural hierarchical Thompson sampling algorithm (HierTS) for this class of problems. Our regret bounds hold for many variants of the problems, including when the tasks are solved sequentially or in parallel; and show that the regret decreases with a more informative prior. Our proofs rely on a novel total variance decomposition that can be applied beyond our models. Our theory is complemented by experiments, which show that the hierarchy helps with knowledge sharing among the tasks. This confirms that hierarchical Bayesian bandits are a universal and statistically-efficient tool for learning to act with similar bandit tasks.

We study Thompson sampling (TS) in online decision making, where the uncertain environment is sampled from a mixture distribution. This is relevant in multi-task learning, where a learning agent faces different classes of problems. We incorporate this structure in a natural way by initializing TS with a mixture prior, and call the resulting algorithm MixTS. To analyze MixTS, we develop a novel and general proof technique for analyzing the concentration of mixture distributions. We use it to derive Bayes regret bounds for MixTS in both linear bandits and finite-horizon reinforcement learning (RL). Our regret bounds reflect the structure of the mixture prior, and depend on the number of mixture components and their width. We demonstrate the empirical effectiveness of MixTS in synthetic and real-world experiments.

Mean rewards of actions are often correlated. The form of these correlations may be complex and unknown a priori, such as the preferences of users for recommended products and their categories. To maximize statistical efficiency, it is important to leverage these correlations when learning. We formulate a bandit variant of this problem where the correlations of mean action rewards are represented by a hierarchical Bayesian model with latent variables. Since the hierarchy can have multiple layers, we call it deep. We propose a hierarchical Thompson sampling algorithm (HierTS) for this problem and show how to implement it efficiently for Gaussian hierarchies. The efficient implementation is possible due to a novel exact hierarchical representation of the posterior, which itself is of independent interest. We use this exact posterior to analyze the Bayes regret of HierTS. Our regret bounds reflect the structure of the problem, that the regret decreases with more informative priors, and can be recast to highlight reduced dependence on the number of actions. We confirm these theoretical findings empirically, in both synthetic and real-world experiments.

We study two model selection settings in stochastic linear bandits (LB). In the first setting, which we refer to as feature selection, the expected reward of the LB problem is in the linear span of at least one of $M$ feature maps (models). In the second setting, the reward parameter of the LB problem is arbitrarily selected from $M$ models represented as (possibly) overlapping balls in $\mathbb R^d$. However, the agent only has access to misspecified models, i.e., estimates of the centers and radii of the balls. We refer to this setting as parameter selection. For each setting, we develop and analyze a computationally efficient algorithm that is based on a reduction from bandits to full-information problems. This allows us to obtain regret bounds that are not worse (up to a $\sqrt{\log M}$ factor) than the case where the true model is known. This is the best reported dependence on the number of models $M$ in these settings. Finally, we empirically show the effectiveness of our algorithms using synthetic and real-world experiments.

Best-arm identification (BAI) in a fixed-budget setting is a bandit problem where the learning agent maximizes the probability of identifying the optimal (best) arm after a fixed number of observations. Most works on this topic study unstructured problems with a small number of arms, which limits their applicability. We propose a general tractable algorithm that incorporates the structure, by successively eliminating suboptimal arms based on their mean reward estimates from a joint generalization model. We analyze our algorithm in linear and generalized linear models (GLMs), and propose a practical implementation based on a G-optimal design. In linear models, our algorithm has competitive error guarantees to prior works and performs at least as well empirically. In GLMs, this is the first practical algorithm with analysis for fixed-budget BAI.

In this work we investigate meta-learning (or learning-to-learn) approaches in multi-task linear stochastic bandit problems that can originate from multiple environments. Inspired by the work of [1] on meta-learning in a sequence of linear bandit problems whose parameters are sampled from a single distribution (i.e., a single environment), here we consider the feasibility of meta-learning when task parameters are drawn from a mixture distribution instead. For this problem, we propose a regularized version of the OFUL algorithm that, when trained on tasks with labeled environments, achieves low regret on a new task without requiring knowledge of the environment from which the new task originates. Specifically, our regret bound for the new algorithm captures the effect of environment misclassification and highlights the benefits over learning each task separately or meta-learning without recognition of the distinct mixture components.

Modern statistical estimation is often performed in a distributed setting where each sample belongs to single user who shares their data with a central server. Users are typically concerned with preserving the privacy of their sample, and also with minimizing the amount of data they must transmit to the server. We give improved private and communication-efficient algorithms for estimating several popular measures of the entropy of a distribution. All of our algorithms have constant communication cost and satisfy local differential privacy. For a joint distribution on many variables whose conditional independence graph is a tree, we describe algorithms for estimating Shannon entropy that require a number of samples that is linear in the number of variables, compared to the quadratic sample complexity of prior work. We also describe an algorithm for estimating Gini entropy whose sample complexity has no dependence on the support size of the distribution and can be implemented using a single round of concurrent communication between the users and the server, while the previously best-known algorithm has high communication cost and requires the server to facilitate interaction between the users. Finally, we describe an algorithm for estimating collision entropy that matches the space and sample complexity of the best known algorithm but generalizes it to the private and communication-efficient setting.

In risk-averse reinforcement learning (RL), the goal is to optimize some risk measure of the returns. A risk measure often focuses on the worst returns out of the agent's experience. As a result, standard methods for risk-averse RL often ignore high-return strategies. We prove that under certain conditions this inevitably leads to a local-optimum barrier, and propose a mechanism we call soft risk to bypass it. We also devise a novel cross entropy module for sampling, which (1) preserves risk aversion despite the soft risk; (2) independently improves sample efficiency. By separating the risk aversion of the sampler and the optimizer, we can sample episodes with poor conditions, yet optimize with respect to successful strategies. We combine these two concepts in CeSoR - Cross-entropy Soft-Risk optimization algorithm - which can be applied on top of any risk-averse policy gradient (PG) method. We demonstrate improved risk aversion in maze navigation, autonomous driving, and resource allocation benchmarks, including in scenarios where standard risk-averse PG completely fails.

The goal of robust reinforcement learning (RL) is to learn a policy that is robust against the uncertainty in model parameters. Parameter uncertainty commonly occurs in many real-world RL applications due to simulator modeling errors, changes in the real-world system dynamics over time, and adversarial disturbances. Robust RL is typically formulated as a max-min problem, where the objective is to learn the policy that maximizes the value against the worst possible models that lie in an uncertainty set. In this work, we propose a robust RL algorithm called Robust Fitted Q-Iteration (RFQI), which uses only an offline dataset to learn the optimal robust policy. Robust RL with offline data is significantly more challenging than its non-robust counterpart because of the minimization over all models present in the robust Bellman operator. This poses challenges in offline data collection, optimization over the models, and unbiased estimation. In this work, we propose a systematic approach to overcome these challenges, resulting in our RFQI algorithm. We prove that RFQI learns a near-optimal robust policy under standard assumptions and demonstrate its superior performance on standard benchmark problems.

We introduce new planning and reinforcement learning algorithms for discounted MDPs that utilize an approximate model of the environment to accelerate the convergence of the value function. Inspired by the splitting approach in numerical linear algebra, we introduce \emph{Operator Splitting Value Iteration} (OS-VI) for both Policy Evaluation and Control problems. OS-VI achieves a much faster convergence rate when the model is accurate enough. We also introduce a sample-based version of the algorithm called OS-Dyna. Unlike the traditional Dyna architecture, OS-Dyna still converges to the correct value function in presence of model approximation error.

We study a constrained contextual linear bandit setting, where the goal of the agent is to produce a sequence of policies, whose expected cumulative reward over the course of multiple rounds is maximum, and each one of them has an expected cost below a certain threshold. We propose an upper-confidence bound algorithm for this problem, called optimistic pessimistic linear bandit (OPLB), and prove a sublinear bound on its regret that is inversely proportional to the difference between the constraint threshold and the cost of a known feasible action. Our algorithm balances exploration and constraint satisfaction using a novel idea that scales the radii of the reward and cost confidence sets with different scaling factors. We further specialize our results to multi-armed bandits and propose a computationally efficient algorithm for this setting and prove a a regret bound that is better than simply casting multi-armed bandits as an instance of linear bandits and using the regret bound of OPLB. We also prove a lower-bound for the problem studied in the paper and provide simulations to validate our theoretical results. Finally, we show how our algorithm and analysis can be extended to multiple constraints and to the case when the cost of the feasible action is unknown.

The convergence rate of Value Iteration (VI), a fundamental procedure in dynamic programming and reinforcement learning, for solving MDPs can be slow when the discount factor is close to one. We propose modifications to VI in order to potentially accelerate its convergence behaviour. The key insight is the realization that the evolution of the value function approximations $(V_k)_{k \geq 0}$ in the VI procedure can be seen as a dynamical system. This opens up the possibility of using techniques from \emph{control theory} to modify, and potentially accelerate, this dynamics. We present such modifications based on simple controllers, such as PD (Proportional-Derivative), PI (Proportional-Integral), and PID. We present the error dynamics of these variants of VI, and provably (for certain classes of MDPs) and empirically (for more general classes) show that the convergence rate can be significantly improved. We also propose a gain adaptation mechanism in order to automatically select the controller gains, and empirically show the effectiveness of this procedure.

Model-based reinforcement learning (RL) algorithms allow us to combine model-generated data with those collected from interaction with the real system in order to alleviate the data efficiency problem in RL. However, designing such algorithms is often challenging because the bias in simulated data may overshadow the ease of data generation. A potential solution to this challenge is to jointly learn and improve model and policy using a universal objective function. In this paper, we leverage the connection between RL and probabilistic inference, and formulate such an objective function as a variational lower-bound of a log-likelihood. This allows us to use expectation maximization (EM) and iteratively fix a baseline policy and learn a variational distribution, consisting of a model and a policy (E-step), followed by improving the baseline policy given the learned variational distribution (M-step). We propose model-based and model-free policy iteration (actor-critic) style algorithms for the E-step and show how the variational distribution learned by them can be used to optimize the M-step in a fully model-based fashion. Our experiments on a number of continuous control tasks show that our model-based (E-step) algorithm, called variational model-based policy optimization (VMBPO), is more sample-efficient and robust to hyper-parameter tuning than its model-free (E-step) counterpart. Using the same control tasks, we also compare VMBPO with several state-of-the-art model-based and model-free RL algorithms and show its sample efficiency and performance.

Machine learning models trained on uncurated datasets can often end up adversely affecting inputs belonging to underrepresented groups. To address this issue, we consider the problem of adaptively constructing training sets which allow us to learn classifiers that are fair in a {\em minimax} sense. We first propose an adaptive sampling algorithm based on the principle of \emph{optimism}, and derive theoretical bounds on its performance. We also propose heuristic extensions of this algorithm suitable for application to large scale, practical problems. Next, by deriving algorithm independent lower-bounds for a specific class of problems, we show that the performance achieved by our adaptive scheme cannot be improved in general. We then validate the benefits of adaptively constructing training sets via experiments on synthetic tasks with logistic regression classifiers, as well as on several real-world tasks using convolutional neural networks (CNNs).

In many fields such as digital marketing, healthcare, finance, and robotics, it is common to have a well-tested and reliable baseline policy running in production (e.g., a recommender system). Nonetheless, the baseline policy is often suboptimal. In this case, it is desirable to deploy online learning algorithms (e.g., a multi-armed bandit algorithm) that interact with the system to learn a better/optimal policy under the constraint that during the learning process the performance is almost never worse than the performance of the baseline itself. In this paper, we study the conservative learning problem in the contextual linear bandit setting and introduce a novel algorithm, the Conservative Constrained LinUCB (CLUCB2). We derive regret bounds for CLUCB2 that match existing results and empirically show that it outperforms state-of-the-art conservative bandit algorithms in a number of synthetic and real-world problems. Finally, we consider a more realistic constraint where the performance is verified only at predefined checkpoints (instead of at every step) and show how this relaxed constraint favorably impacts the regret and empirical performance of CLUCB2.

While learning in an unknown Markov Decision Process (MDP), an agent should trade off exploration to discover new information about the MDP, and exploitation of the current knowledge to maximize the reward. Although the agent will eventually learn a good or optimal policy, there is no guarantee on the quality of the intermediate policies. This lack of control is undesired in real-world applications where a minimum requirement is that the executed policies are guaranteed to perform at least as well as an existing baseline. In this paper, we introduce the notion of conservative exploration for average reward and finite horizon problems. We present two optimistic algorithms that guarantee (w.h.p.) that the conservative constraint is never violated during learning. We derive regret bounds showing that being conservative does not hinder the learning ability of these algorithms.

We study two randomized algorithms for generalized linear bandits. The first, GLM-TSL, samples a generalized linear model (GLM) from the Laplace approximation to the posterior distribution. The second, GLM-FPL, fits a GLM to a randomly perturbed history of past rewards. We analyze both algorithms and derive $\tilde{O}(d \sqrt{n \log K})$ upper bounds on their $n$-round regret, where $d$ is the number of features and $K$ is the number of arms. The former improves on prior work while the latter is the first for Gaussian noise perturbations in non-linear models. We empirically evaluate both GLM-TSL and GLM-FPL in logistic bandits, and apply GLM-FPL to neural network bandits. Our work showcases the role of randomization, beyond posterior sampling, in exploration.

We consider the problem of allocating a fixed budget of samples to a finite set of discrete distributions to learn them uniformly well (minimizing the maximum error) in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general \emph{optimistic tracking algorithm} and analyze its sample allocation performance w.r.t. an oracle. We then instantiate this algorithm for the four distance measures and derive bounds on their regret. We also show that the allocation performance of the proposed algorithm cannot, in general, be improved, by deriving lower-bounds on the expected deviation from the oracle allocation for any adaptive scheme. We verify our theoretical findings through some experiments. Finally, we show that the techniques developed in the paper can be easily extended to learn some classes of continuous distributions as well as to the related setting of minimizing the average error (in terms of the four distances) in learning a set of distributions.

High-dimensional observations and unknown dynamics are major challenges when applying optimal control to many real-world decision making tasks. The Learning Controllable Embedding (LCE) framework addresses these challenges by embedding the observations into a lower dimensional latent space, estimating the latent dynamics, and then performing control directly in the latent space. To ensure the learned latent dynamics are predictive of next-observations, all existing LCE approaches decode back into the observation space and explicitly perform next-observation prediction—a challenging high-dimensional task that furthermore introduces a large number of nuisance parameters (i.e., the decoder) which are discarded during control. In this paper, we propose a novel information-theoretic LCE approach and show theoretically that explicit next-observation prediction can be replaced with predictive coding. We then use predictive coding to develop a decoder-free LCE model whose latent dynamics are amenable to locally-linear control. Extensive experiments on benchmark tasks show that our model reliably learns a controllable latent space that leads to superior performance when compared with state-of-the-art LCE baselines.

Multi-step greedy policies have been extensively used in model-based reinforcement learning (RL), both when a model of the environment is available (e.g., in the game of Go) and when it is learned. In this paper, we explore their benefits in model-free RL, when employed using multi-step dynamic programming algorithms: $\kappa$-Policy Iteration ($\kappa$-PI) and $\kappa$-Value Iteration ($\kappa$-VI). These methods iteratively compute the next policy ($\kappa$-PI) and value function ($\kappa$-VI) by solving a surrogate decision problem with a shaped reward and a smaller discount factor. We derive model-free RL algorithms based on $\kappa$-PI and $\kappa$-VI in which the surrogate problem can be solved by any discrete or continuous action RL method, such as DQN and TRPO. We identify the importance of a hyper-parameter that controls the extent to which the surrogate problem is solved and suggest a way to set this parameter. When evaluated on a range of Atari and MuJoCo benchmark tasks, our results indicate that for the right range of $\kappa$, our algorithms outperform DQN and TRPO. This shows that our multi-step greedy algorithms are general enough to be applied over any existing RL algorithm and can significantly improve its performance.

We consider the problem of binary classification with the caveat that the learner can abstain from declaring a label incurring a cost λ ∈ [0,1/2] in the process. This is referred to as the problem of binary classification with a fixed-cost of abstention. For this problem, we propose an active learning strategy that constructs a non-uniform partition of the input space and focuses sampling in the regions near the decision boundaries. Our proposed algorithm can work in all the commonly used active learning query models, namely membership-query, pool-based and stream-based. We obtain an upper bound on the excess risk of our proposed algorithm under standard smoothness and margin assumptions and demonstrate its minimax near-optimality by deriving a matching (modulo poly-logarithmic factors) lower bound. The achieved minimax rates are always faster than the corresponding rates in the passive setting, and furthermore the improvement increases with larger values of the smoothness and margin parameters.

Real Time Dynamic Programming (RTDP) is an online algorithm based on Dynamic Programming (DP) that acts by 1-step greedy planning. Unlike DP, RTDP does not require access to the entire state space, i.e., it explicitly handles the exploration. This fact makes RTDP particularly appealing when the state space is large and it is not possible to update all states simultaneously. In this we devise a multi-step greedy RTDP algorithm, which we call $h$-RTDP, that replaces the 1-step greedy policy with a $h$-step lookahead policy. We analyze $h$-RTDP in its exact form and establish that increasing the lookahead horizon, $h$, results in an improved sample complexity, with the cost of additional computations. This is the first work that proves improved sample complexity as a result of {\em increasing} the lookahead horizon in online planning. We then analyze the performance of $h$-RTDP in three approximate settings: approximate model, approximate value updates, and approximate state representation. For these cases, we prove that the asymptotic performance of $h$-RTDP remains the same as that of a corresponding approximate DP algorithm, the best one can hope for without further assumptions on the approximation errors.

We study the problem of efficient exploration in order to learn an accurate model of an environment, modeled as a Markov decision process (MDP). Efficient exploration in this problem requires the agent to identify the regions in which estimating the model is more difficult and then exploit this knowledge to collect more samples there. In this paper, we formalize this problem, introduce the first algorithm to learn an $\epsilon$-accurate estimate of the dynamics, and provide its sample complexity analysis. While this algorithm enjoys strong guarantees in the large-sample regime, it tends to have a poor performance in early stages of exploration. To address this issue, we propose an algorithm that is based on maximum weighted entropy, a heuristic that stems from common sense and our theoretical analysis. The main idea here is to cover the entire state-action space with the weight proportional to the noise in their transition functions. Using a number of simple domains with heterogeneous noise in their transitions, we show that our heuristic-based algorithm outperforms both our original algorithm and the maximum entropy algorithm in the small sample regime, while achieving similar asymptotic performance as that of the original algorithm.

We address the problem of finding an optimal policy in a Markov decision process (MDP) under a restricted policy class defined by the convex hull of a set of base policies. This problem is of great interest in applications in which a number of reasonably good (or safe) policies are already known and we are interested in optimizing in their convex hull. We first prove that solving this problem is NP-hard. We then propose an efficient algorithm that finds a policy whose performance is almost as good as that of the best convex combination of the base policies, under the assumption that the occupancy measures of the base policies have a large overlap. The running time of the proposed algorithm is linear in the number of states and polynomial in the number of base policies. A distinct advantage of the proposed algorithm is that, apart from the computation of the occupancy measures of the base policies, it does not need to interact with the environment during the optimization process. This is especially important (i) in problems that due to concerns such as safety, we are restricted in interacting with the environment only through the (safe) base policies, and (ii) in complex systems where estimating the value of a policy can be a time consuming process.

We study risk-sensitive imitation learning where the agent’s goal is to perform at least as well as the expert in terms of a risk profile. We first formulate our risk-sensitive imitation learning setting. We consider the generative adversarial approach to imitation learning (GAIL) and derive an optimization problem for our formulation, which we call it risk- sensitive GAIL (RS-GAIL). We then derive two different versions of our RS-GAIL optimization problem that aim at matching the risk profiles of the agent and the expert w.r.t. Jensen-Shannon (JS) divergence and Wasserstein distance, and develop risk-sensitive generative adversarial imitation learning algorithms based on these optimization problems. We evaluate the performance of our algorithms and compare them with GAIL and the risk-averse imitation learning (RAIL) algorithms in two MuJoCo and two OpenAI classical control tasks.

We propose a bandit algorithm that explores by randomizing its history of rewards. Specifically, it pulls the arm with the highest mean reward in a non-parametric bootstrap sample of its history with pseudo rewards. We design the pseudo rewards such that the bootstrap mean is optimistic with a sufficiently high probability. We call our algorithm Giro, which stands for garbage in, reward out. We analyze Giro in a Bernoulli bandit and derive a $O(K \Delta^{-1} \log n)$ bound on its $n$-round regret, where $\Delta$ is the difference in the expected rewards of the optimal and the best suboptimal arms, and $K$ is the number of arms. The main advantage of our exploration design is that it easily generalizes to structured problems. To show this, we propose contextual Giro with an arbitrary reward generalization model. We evaluate Giro and its contextual variant on multiple synthetic and real-world problems, and observe that it performs well.

We propose an online algorithm for cumulative regret minimization in a stochastic multi-armed bandit. The algorithm adds O(t) i.i.d. pseudo-rewards to its history in round t and then pulls the arm with the highest average reward in its perturbed history. Therefore, we call it perturbed-history exploration (PHE). The pseudo-rewards are carefully designed to offset potentially underestimated mean rewards of arms with a high probability. We derive near-optimal gap-dependent and gap-free bounds on the n-round regret of PHE. The key step in our analysis is a novel argument that shows that randomized Bernoulli rewards lead to optimism. Finally, we empirically evaluate PHE and show that it is competitive with state-of-the-art baselines.

State-of-the-art efficient model-based Reinforcement Learning (RL) algorithms typically act by iteratively solving empirical models, i.e., by performing full-planning on Markov Decision Processes (MDPs) built by the gathered experience. In this paper, we focus on model-based RL in the finite-state finite-horizon MDP setting and establish that exploring with greedy policies -- act by 1-step planning -- can achieve tight minimax performance in terms of regret, O(\sqrt{HSAT}). Thus, full-planning in model-based RL can be avoided altogether without any performance degradation, and, by doing so, the computational complexity decreases by a factor of S. The results are based on a novel analysis of real-time dynamic programming, then extended to model-based RL. Specifically, we generalize existing algorithms that perform full-planning to such that act by 1-step planning. For these generalizations, we prove regret bounds with the same rate as their full-planning counterparts.

Embed-to-control (E2C) is a model for solving high-dimensional optimal control problems by combining variational auto-encoders with locally-optimal controllers. However, the E2C model suffers from two major drawbacks: 1) its objective function does not correspond to the likelihood of the data sequence and 2) the variational encoder used for embedding typically has large variational approximation error, especially when there is noise in the system dynamics. In this paper, we present a new model for learning robust locally-linear controllable embedding (RCE). Our model directly estimates the predictive conditional density of the future observation given the current one, while introducing the bottleneck between the current and future observations. Although the bottleneck provides a natural embedding candidate for control, our RCE model introduces additional specific structures in the generative graphical model so that the model dynamics can be robustly linearized. We also propose a principled variational approximation of the embedding posterior that takes the future observation into account, and thus, makes the variational approximation more robust against the noise. Experimental results show that RCE outperforms the E2C model, and does so significantly when the underlying dynamics is noisy.

We study the sparse entropy-regularized reinforcement learning (ERL) problem in which the entropy term is a special form of the Tsallis entropy. The optimal policy of this formulation is sparse, i.e., at each state, it has non-zero probability for only a small number of actions. This addresses the main drawback of the standard Shannon entropy-regularized RL (soft ERL) formulation, in which the optimal policy is softmax, and thus, may assign a non-negligible probability mass to non-optimal actions. This problem is aggravated as the number of actions is increased. In this paper, we follow the work of Nachum et al. (2017) in the soft ERL setting, and propose a class of novel path consistency learning (PCL) algorithms, called sparse PCL, for the sparse ERL problem that can work with both on-policy and off-policy data. We first derive a sparse consistency equation that specifies a relationship between the optimal value function and policy of the sparse ERL along any system trajectory. Crucially, a weak form of the converse is also true, and we quantify the sub-optimality of a policy which satisfies sparse consistency, and show that as we increase the number of actions, this sub-optimality is better than that of the soft ERL optimal policy. We then use this result to derive the sparse PCL algorithms. We empirically compare sparse PCL with its soft counterpart, and show its advantage, especially in problems with a large number of actions.

We study the problem of off-policy evaluation (OPE) in reinforcement learning (RL), where the goal is to estimate the performance of a policy from the data generated by another policy(ies). In particular, we focus on the doubly robust (DR) estimators that consist of an importance sampling (IS) component and a performance model, and utilize the low (or zero) bias of IS and low variance of the model at the same time. Although the accuracy of the model has a huge impact on the overall performance of DR, most of the work on using the DR estimators in OPE has been focused on improving the IS part, and not much on how to learn the model. In this paper, we propose alternative DR estimators, called more robust doubly robust (MRDR), that learn the model parameter by minimizing the variance of the DR estimator. We first present a formulation for learning the DR model in RL. We then derive formulas for the variance of the DR estimator in both contextual bandits and RL, such that their gradients w.r.t. the model parameters can be estimated from the samples, and propose methods to efficiently minimize the variance. We prove that the MRDR estimators are strongly consistent and asymptotically optimal. Finally, we evaluate MRDR in bandits and RL benchmark problems, and compare its performance with the existing methods.

In many real-world reinforcement learning (RL) problems, besides optimizing the main objective function, an agent must concurrently avoid violating a number of constraints. In particular, besides optimizing performance, it is crucial to guarantee the safety of an agent during training as well as deployment (e.g., a robot should avoid taking actions - exploratory or not - which irrevocably harm its hard- ware). To incorporate safety in RL, we derive algorithms under the framework of constrained Markov decision processes (CMDPs), an extension of the standard Markov decision processes (MDPs) augmented with constraints on expected cumulative costs. Our approach hinges on a novel Lyapunov method. We define and present a method for constructing Lyapunov functions, which provide an effective way to guarantee the global safety of a behavior policy during training via a set of local linear constraints. Leveraging these theoretical underpinnings, we show how to use the Lyapunov approach to systematically transform dynamic programming (DP) and RL algorithms into their safe counterparts. To illustrate their effectiveness, we evaluate these algorithms in several CMDP planning and decision-making tasks on a safety benchmark domain. Our results show that our proposed method significantly outperforms existing baselines in balancing constraint satisfaction and performance.

Risk management in dynamic decision problems is a primary concern in many fields, including financial investment, autonomous driving, and healthcare. The mean-variance function is one of the most widely used objective functions in risk management due to its simplicity and interpretability. Existing algorithms for mean-variance optimization are based on multi-time-scale stochastic approximation, whose learning rate schedules are often hard to tune, and have only asymptotic convergence proof. In this paper, we develop a model-free policy search framework for mean-variance optimization with finite-sample error bound analysis (to local optima). Our starting point is a reformulation of the original mean-variance function with its Fenchel dual, from which we propose a stochastic block coordinate ascent policy search algorithm. Both the asymptotic convergence guarantee of the last iteration's solution and the convergence rate of the randomly picked solution are provided, and their applicability is demonstrated on several benchmark domains.

This work studies multiple hypothesis testing in the setting when we obtain data sequentially and may choose when to stop sampling. We summarize the notion of a sequential p-value (one that can be continually updated and still maintain a type I error guarantee) and provide several examples from the literature. This tool allows us to convert step-up or step-down multiple hypothesis testing procedures in the fixed-horizon setting (which includes Benjamini-Hochberg, Holm, and Bonferroni) into sequential versions that allow the statistician to reject a hypothesis as soon as the sequential p-value reaches a threshold. We show that if the original procedure has a type I error guarantee in a certain family (including FDR and FWER), then the sequential conversion inherits an analogous guarantee. The conversion also allows for allocating samples in a data-dependent way, and we provide simulated experiments demonstrating an increased number of rejections when compared to the fixed-horizon setting.

We explore the sequential decision making problem where the goal is to estimate uniformly well a number of linear models, given a shared budget of random contexts independently sampled from a known distribution. The decision maker must query one of the linear models for each incoming context, and receives an observation corrupted by noise levels that are unknown, and depend on the model instance. We present Trace-UCB, an adaptive allocation algorithm that learns the noise levels while balancing contexts accordingly across the different linear functions, and derive guarantees for simple regret in both expectation and high-probability. Finally, we extend the algorithm and its guarantees to high dimensional settings, where the number of linear models times the dimension of the contextual space is higher than the total budget of samples. Simulations with real data suggest that Trace-UCB is remarkably robust, outperforming a number of baselines even when its assumptions are violated.

We introduce a new framework for training deep generative models for high-dimensional conditional density estimation. The Bottleneck Conditional Density Estimator (BCDE) is a variant of the conditional variational autoencoder (CVAE) that employs layer(s) of stochastic variables as the bottleneck between the input x and target y, where both are high-dimensional. Crucially, we propose a new hybrid training method that blends the conditional generative model with a joint generative model. Hybrid blending is the key to effective training of the BCDE, which avoids overfitting and provides a novel mechanism for leveraging unlabeled data. We show that our hybrid training procedure enables models to achieve competitive results in the MNIST quadrant prediction task in the fully-supervised setting, and sets new benchmarks in the semi-supervised regime for MNIST, SVHN, and CelebA.

We consider influence maximization (IM) in social networks, which is the problem of maximizing the number of users that become aware of a product by selecting a set of “seed” users to expose the product to. While prior work assumes a known model of information diffusion, we propose a novel parametrization that not only makes our framework agnostic to the underlying diffusion model, but also statistically efficient to learn from data. We give a corresponding monotone, submodular surrogate function, and show that it is a good approximation to the original IM objective. We also consider the case of a new marketer looking to exploit an existing social network, while simultaneously learning the factors governing information propagation. For this, we propose a pairwise-influence semi-bandit feedback model and develop a LinUCB-based bandit algorithm. Our model-independent analysis shows that our regret bound has a better (as compared to previous work) dependence on the size of the network. Experimental evaluation suggests that our framework is robust to the underlying diffusion model and can efficiently learn a near-optimal solution.

Online learning to rank is a core problem in information retrieval and machine learning. Many provably efficient algorithms have been recently proposed for this problem in specific click models. The click model is a model of how the user interacts with a list of documents. Though these results are significant, their impact on practice is limited, because all proposed algorithms are designed for specific click models and lack convergence guarantees in other models. In this work, we propose BatchRank, the first online learning to rank algorithm for a broad class of click models. The class encompasses two most fundamental click models, the cascade and position-based models. We derive a gap-dependent upper bound on the T-step regret of BatchRank and evaluate it on a range of web search queries. We observe that BatchRank outperforms ranked bandits and is more robust than CascadeKL-UCB, an existing algorithm for the cascade model.

Safety is a desirable property that can immensely increase the applicability of learning algorithms in real-world decision-making problems. It is much easier for a company to deploy an algorithm that is safe, i.e., guaranteed to perform at least as well as a baseline. In this paper, we study the issue of safety in contextual linear bandits that have application in many different fields including personalized ad recommendation in online marketing. We formulate a notion of safety for this class of algorithms. We develop a safe contextual linear bandit algorithm, called conservative linear UCB (CLUCB), that simultaneously minimizes its regret and satisfies the safety constraint, i.e., maintains its performance above a fixed percentage of the performance of a baseline strategy, uniformly over time. We prove an upper-bound on the regret of CLUCB and show that it can be decomposed into two terms: 1) an upper-bound for the regret of the standard linear UCB algorithm that grows with the time horizon and 2) a constant term that accounts for the loss of being conservative in order to satisfy the safety constraint. We empirically show that our algorithm is safe and validate our theoretical analysis.

We study the problem of combinatorial pure exploration in the stochastic multi-armed bandit problem. We first construct a new measure of complexity that provably characterizes the learning performance of the algorithms we propose for the fixed confidence and the fixed budget setting. We show that this complexity is never higher than the one in existing work and illustrate a number of configurations in which it can be significantly smaller. While in general this improvement comes at the cost of increased computational complexity, we provide a series of examples, including a planning problem, where this extra cost is not significant.

An important problem in sequential decision-making under uncertainty is to use limited data to compute a safe policy, i.e., a policy that is guaranteed to perform at least as well as a given baseline strategy. In this paper, we develop and analyze a new model-based approach to compute a safe policy when we have access to an inaccurate dynamics model of the system with known accuracy guarantees. Our proposed robust method uses this (inaccurate) model to directly minimize the (negative) regret w.r.t. the baseline policy. Contrary to the existing approaches, minimizing the regret allows one to improve the baseline policy in states with accurate dynamics and seamlessly fall back to the baseline policy, otherwise. We show that our formulation is NP-hard and propose an approximate algorithm. Our empirical results on several domains show that even this relatively simple approximate algorithm can significantly outperform standard approaches.

We study two regularization-based approximate policy iteration algorithms, namely REG-LSPI and REG-BRM, to solve reinforcement learning and planning problems in discounted Markov Decision Processes with large state and finite action spaces. The core of these algorithms are the regularized extensions of the Least- Squares Temporal Difference (LSTD) learning and Bellman Residual Minimization (BRM), which are used in the algorithms' policy evaluation steps. Regularization provides a convenient way to control the complexity of the function space to which the estimated value function belongs and as a result enables us to work with rich nonparametric function spaces. We derive efficient implementations of our methods when the function space is a reproducing kernel Hilbert space. We analyze the statistical properties of REG-LSPI and provide an upper bound on the policy evaluation error and the performance loss of the policy returned by this method. Our bound shows the dependence of the loss on the number of samples, the capacity of the function space, and some intrinsic properties of the underlying Markov Decision Process. The dependence of the policy evaluation bound on the number of samples is minimax optimal. This is the first work that provides such a strong guarantee for a nonparametric approximate policy iteration algorithm. (This work is an extension of the NIPS 2008 conference paper by Farahmand et al. (2009b).)

Policy gradient methods are reinforcement learning algorithms that adapt a parameterized policy by following a performance gradient estimate. Many conventional policy gradient methods use Monte-Carlo techniques to estimate this gradient. The policy is improved by adjusting the parameters in the direction of the gradient estimate. Since Monte-Carlo methods tend to have high variance, a large number of samples is required to attain accurate estimates, resulting in slow convergence. In this paper, we first propose a Bayesian framework for policy gradient, based on modeling the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient as well as a measure of the uncertainty in the gradient estimates, namely, the gradient covariance, are provided at little extra cost. Since the proposed Bayesian framework considers system trajectories as its basic observable unit, it does not require the dynamics within trajectories to be of any particular form, and thus, can be easily extended to partially observable problems. On the downside, it cannot take advantage of the Markov property when the system is Markovian. To address this issue, we proceed to supplement our Bayesian policy gradient framework with a new actor-critic learning model in which a Bayesian class of non- parametric critics, based on Gaussian process temporal difference learning, is used. Such critics model the action- value function as a Gaussian process, allowing Bayes' rule to be used in computing the posterior distribution over action-value functions, conditioned on the observed data. Appropriate choices of the policy parameterization and of the prior covariance (kernel) between action-values allow us to obtain closed-form expressions for the posterior distribution of the gradient of the expected return with respect to the policy parameters. We perform detailed experimental comparisons of the proposed Bayesian policy gradient and actor-critic algorithms with classic Monte-Carlo based policy gradient methods, as well as with each other, on a number of reinforcement learning problems.

We introduce a variant of the classification-based approach to policy iteration which uses a cost-sensitive loss function weighting each classification mistake by its actual regret, that is, the difference between the action- value of the greedy action and of the action chosen by the classifier. For this algorithm, we provide a full finite-sample analysis. Our results state a performance bound in terms of the number of policy improvement steps, the number of rollouts used in each iteration, the capacity of the considered policy space (classifier), and a capacity measure which indicates how well the policy space can approximate policies that are greedy with respect to any of its members. The analysis reveals a tradeoff between the estimation and approximation errors in this classification-based policy iteration setting. Furthermore it confirms the intuition that classification-based policy iteration algorithms could be favorably compared to value-based approaches when the policies can be approximated more easily than their corresponding value functions. We also study the consistency of the algorithm when there exists a sequence of policy spaces with increasing capacity.

Many reinforcement learning algorithms use trajectories collected from the execution of one or more policies to propose a new policy. Because execution of a bad policy can be costly or dangerous, techniques for evaluating the performance of the new policy without requiring its execution have been of recent interest in industry. Such off-policy evaluation methods, which estimate the performance of a policy using trajectories collected from the execution of other policies, heretofore have not provided confidences regarding the accuracy of their estimates. In this paper we propose an off-policy method for computing a lower confidence bound on the expected return of a policy.

We present a batch reinforcement learning (RL) algorithm that provides probabilistic guarantees about the quality of each policy that it proposes, and which has no hyper-parameter that requires expert tuning. Specifically, the user may select any performance lower-bound and confidence level and our algorithm will ensure that the probability that it returns a policy with performance below the lower bound is at most the specified confidence level. We then propose an incremental algorithm that executes our policy improvement algorithm repeatedly to generate multiple policy improvements. We show the viability of our approach with a simple 4 x 4 gridworld and the standard mountain car problem, as well as with a digital marketing application that uses real world data.

Several authors have recently developed risk-sensitive policy gradient methods that augment the standard expected cost minimization problem with a measure of variability in cost. These studies have focused on specific risk-measures, such as the variance or conditional value at risk (CVaR). In this work, we extend the policy gradient method to the whole class of coherent risk measures, which is widely accepted in finance and operations research, among other fields. We consider both static and time-consistent dynamic risk measures. For static risk measures, our approach is in the spirit of policy gradient algorithms and combines a standard sampling approach with convex programming. For dynamic risk measures, our approach is actor-critic style and involves explicit approximation of value function. Most importantly, our contribution presents a unified approach to risk-sensitive reinforcement learning that generalizes and extends previous results.

We study the problem of structured output learning from a regression perspective. We first provide a general formulation of the kernel dependency estimation (KDE) approach to this problem using operator-valued kernels. Our formulation overcomes the two main limitations of the original KDE approach, namely the decoupling between outputs in the image space and the inability to use a joint feature space. We then propose a covariance-based operator-valued kernel that allows us to take into account the structure of the kernel feature space. This kernel operates on the output space and only encodes the interactions between the outputs without any reference to the input space. To address this issue, we introduce a variant of our KDE method based on the conditional covariance operator that in addition to the correlation between the outputs takes into account the effects of the input variables. Finally, we evaluate the performance of our KDE approach using both covariance and conditional covariance kernels on three structured output problems, and compare it to the state-of-the art kernel-based structured output regression methods.

A commonly used approach to multiclass classification is to replace the 0-1 loss with a convex surrogate so as to make empirical risk minimization computationally tractable. Previous work has uncovered sufficient and necessary conditions for the consistency of the resulting procedures. In this paper, we strengthen these results by showing how the 0-1 excess loss of a predictor can be upper bounded as a function of the excess loss of the predictor measured using the convex surrogate. The bound is developed for the case of cost-sensitive multiclass classification and a convex surrogate loss that goes back to the work of Lee, Lin and Wahba. The bounds are as easy to calculate as in binary classification. Furthermore, we also show that our analysis extends to the analysis of the recently introduced “Simplex Coding” scheme.

Tetris is a popular video game that has been widely used as a benchmark for various optimization techniques including approximate dynamic programming (ADP) algorithms. A close look at the literature of this game shows that while ADP algorithms, that have been (almost) entirely based on approximating the value function (value function based), have performed poorly in Tetris, the methods that search directly in the space of policies by learning the policy parameters using an optimization black box, such as the cross entropy (CE) method, have achieved the best reported results. This makes us conjecture that Tetris is a game in which good policies are easier to represent, and thus, learn than their corresponding value functions. So, in order to obtain a good performance with ADP, we should use ADP algorithms that search in a policy space, instead of the more traditional ones that search in a value function space. In this paper, we put our conjecture to test by applying such an ADP algorithm, called classification-based modified policy iteration (CBMPI), to the game of Tetris. Our extensive experimental results show that for the first time an ADP algorithm, namely CBMPI, obtains the best results reported in the literature for Tetris in both small $10\times 10$ and large $10\times 20$ boards. Although the CBMPI's results are similar to those achieved by the CE method in the large board, CBMPI uses considerably fewer (almost 1/10) samples (call to the generative model of the game) than CE.

In many sequential decision-making problems we may want to manage risk by minimizing some measure of variability in rewards in addition to maximizing a standard criterion. Variance related risk measures are among the most common risk-sensitive criteria in finance and operations research. However, optimizing many such criteria is known to be a hard problem. In this paper, we consider both discounted and average reward Markov decision processes. For each formulation, we first define a measure of variability for a policy, which in turn gives us a set of risk-sensitive criteria to optimize. For each of these criteria, we derive a formula for computing its gradient. We then devise actor-critic algorithms for estimating the gradient and updating the policy parameters in the ascent direction. We establish the convergence of our algorithms to locally risk-sensitive optimal policies. Finally, we demonstrate the usefulness of our algorithms in a traffic signal control application.

The existing classification-based policy iteration (CBPI) algorithms can be divided into two categories: direct policy iteration (DPI) methods that directly assign the output of the classifier (the approximate greedy policy w.r.t.~the current policy) to the next policy, and conservative policy iteration (CPI) methods in which the new policy is a mixture distribution of the current policy and the output of the classifier. The conservative policy update gives CPI a desirable feature, namely the guarantee that the policies generated by this algorithm improve at each iteration. We provide a detailed algorithmic and theoretical comparison of these two classes of CBPI algorithms. Our results reveal that in order to achieve the same level of accuracy, CPI requires more iterations, and thus, more samples than the DPI algorithm. Furthermore, CPI may converge to suboptimal policies whose performance is not better than DPI's.

We study the problem of identifying the best arm(s) in the stochastic multi-armed bandit setting. This problem has been studied in the literature from two different perspectives: fixed budget and fixed confidence. We propose a unifying approach that leads to a meta-algorithm called unified gap-based exploration (UGapE), with a common structure and similar theoretical analysis for these two settings. We prove a performance bound for the two versions of the algorithm showing that the two problems are characterized by the same notion of complexity. We also show how the UGapE algorithm as well as its theoretical analysis can be extended to take into account the variance of the arms and to multiple bandits. Finally, we evaluate the performance of UGapE and compare it with a number of existing fixed budget and fixed confidence algorithms.

We introduce a new convergent variant of Q-learning, called speedy Q-learning, to address the problem of slow convergence in the standard form of the Q-learning algorithm. We prove a PAC bound on the performance of SQL, which shows that for an MDP with n state-action pairs and the discount factor \gamma only T=O\big(\log(n)/(\epsilon^{2}(1-\gamma)^{4})\big) steps are required for the SQL algorithm to converge to an \epsilon-optimal action-value function with high probability. This bound has a better dependency on 1/\epsilon and 1/(1-\gamma), and thus, is tighter than the best available result for Q-learning. Our bound is also superior to the existing results for both model-free and model-based instances of batch Q-value iteration that are considered to be more efficient than the incremental methods like Q-learning.

We study the problem of identifying the best arm in each of the bandits in a multi-bandit multi-armed setting. We first propose an algorithm called Gap-based Exploration (GapE) that focuses on the arms whose mean is close to the mean of the best arm in the same bandit (i.e., small gap). We then introduce an algorithm, called GapE-V, which takes into account the variance of the arms in addition to their gap. We prove an upper-bound on the probability of error for both algorithms. Since GapE and GapE-V need to tune an exploration parameter that depends on the complexity of the problem, which is often unknown in advance, we also introduce variations of these algorithms that estimate this complexity online. Finally, we evaluate the performance of these algorithms and compare them to other allocation strategies on a number of synthetic problems.

We consider the problem of reinforcement learning in high-dimensional spaces when the number of features is bigger than the number of samples. In particular, we study the least-squares temporal difference (LSTD) learning algorithm when a space of low dimension is generated with a random projection from a high-dimensional space. We provide a thorough theoretical analysis of the LSTD with random projections and derive performance bounds for the resulting algorithm. We also show how the error of LSTD with random projections is propagated through the iterations of a policy iteration algorithm and provide a performance bound for the resulting least-squares policy iteration (LSPI) algorithm.

In this paper we consider approximate policy-iteration-based reinforcement learning algorithms. In order to implement a flexible function approximation scheme we propose the use of non-parametric methods with regularization, providing a convenient way to control the complexity of the function approximator. We propose two novel regularized policy iteration algorithms by adding L2-regularization to two widely-used policy evaluation methods: Bellman residual minimization (BRM) and least-squares temporal difference learning (LSTD). We derive efficient implementation for our algorithms when the approximate value-functions belong to a reproducing kernel Hilbert space. We also provide finite-sample performance bounds for our algorithms and show that they are able to achieve optimal rates of convergence under the studied conditions.

We present a new actor-critic learning model in which a Bayesian class of non-parametric critics, using Gaussian process temporal difference learning is used. Such critics model the state-action value function as a Gaussian process, allowing Bayes' rule to be used in computing the posterior distribution over state-action value functions, conditioned on the observed data. Appropriate choices of the prior covariance (kernel) between state-action values and of the parametrization of the policy allow us to obtain closed-form expressions for the posterior distribution of the gradient of the average discounted return with respect to the policy parameters. The posterior mean, which serves as our estimate of the policy gradient, is used to update the policy, while the posterior covariance allows us to gauge the reliability of the update.

We present four new reinforcement learning algorithms based on actor-critic and natural-gradient ideas, and provide their convergence proofs. Actor-critic rein- forcement learning methods are online approximations to policy iteration in which the value-function parameters are estimated using temporal difference learning and the policy parameters are updated by stochastic gradient descent. Methods based on policy gradients in this way are of special interest because of their com- patibility with function approximation methods, which are needed to handle large or in(cid:2)nite state spaces. The use of temporal difference learning in this way is of interest because in many applications it dramatically reduces the variance of the gradient estimates. The use of the natural gradient is of interest because it can produce better conditioned parameterizations and has been shown to further re- duce variance in some cases. Our results extend prior two-timescale convergence results for actor-critic methods by Konda and Tsitsiklis by using temporal differ- ence learning in the actor and by incorporating natural gradients, and they extend prior empirical studies of natural actor-critic methods by Peters, Vijayakumar and Schaal by providing the (cid:2)rst convergence proofs and the (cid:2)rst fully incremental algorithms.

Policy gradient methods are reinforcement learning algorithms that adapt a param- eterized policy by following a performance gradient estimate. Conventional pol- icy gradient methods use Monte-Carlo techniques to estimate this gradient. Since Monte Carlo methods tend to have high variance, a large number of samples is required, resulting in slow convergence. In this paper, we propose a Bayesian framework that models the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient as well as a measure of the uncertainty in the gradient estimates are provided at little extra cost.